Hamiltonian Chess

Introduction

The aim of this chess variant is to be the first to form a
Hamiltonian Path or
Circuit
between all of your pieces. The same piece may not be visited twice in
a path or circuit.

In the diagram below, white has formed the circuit
e2-d1-b3-b8-g3-e4-f6-a6 and then back to e2,
and black has formed the path e3-h6-h7-e7-d5-a5-a3-b1.

Setup

Each player starts with eight pieces (pawns being excluded from the game).
Players alternate placing one of their opponent's pieces on the board
at a time, starting with white placing a black piece.

White moves first.

Rules

There is no notion of Check or Mate in this game, and the King has
no particular disadvantages or privileges (such as castling).

Movement and capturing is performed in the same way as in FIDE Chess.
Note that it may benefit your opponent more than yourself to capture their pieces,
since it makes it simpler for them to form a complete path or circuit. (Note that
a single remaining piece counts as a circuit).

At any time during the game, a player may announce victory by pointing
out a completed Hamiltonian Path or Circuit involving all of their remaining pieces
(starting from any piece).
If the other player is able to point out a path or circuit involving all of their own
pieces, the game is a draw; otherwise the announcing player wins. If a path or
circuit is completed without being announced by the player, victory is not
automatic (and there is no retrospective victory if it is later destroyed).
The only exception to this rule is if a player is reduced to a single piece,
in which case the game ends immediately, just as if the player had announced
victory.

Scoring

If multiple games are played, the winner of each game scores as follows:

One point for each remaining piece (winner's pieces only)

Double points for a circuit

Thus the number of points per game is from 2 to 16 (since a single remaining
piece counts as a circuit), or 0 in the case of a draw.

If a player resigns, their opponent scores twice the number of their
remaining pieces (as if they had completed a circuit).

Notes

It is probably wise to place your opponent's bishops on different colored squares.

An alternative scoring system which discourages this behavior is to score the total length of the path/circuit rather than the number
of remaining pieces, where the distance between two pieces equals the maximum
of the vertical and horizontal difference in position (e.g.. a knight move would
always score 2 points; the maximum is 7). A circuit consisting of a single
piece scores 1 point rather than 0.

The maximum attainable score using this alternative system is left
as an exercise to the reader; the upper bound is (maximum score of 7) * (8 circuit segments) * (2 for being a circuit) = 112.
﻿

As an abstract game fan, I definitely like this game. One that is quite
difficult to quantify. The players placing opponent pieces is a nice
twist.
Although it might be stated that the connectivity of the pieces is based
upon their movement. This is obvious by the example, but the plain text
might give the impression of simple adjacency.